Generation Of Random Processes Research Articles

Simulation of Random Processes in the Fatigue Problem under Variable Loading

The method is proposed to generate random processes with stable integral characteristics of cumulative distributions, which at the same time are random in nature. The algorithm consists of the two steps. The first one is generation of the random extrema sequence using the objective Markovian matrix, completed with in-service loading data. The extrema and their sequence are shown to be of importance for the metal fatigue assessment. Therefore, random testing with extrema control is more advantageous as compared to testing with the generation of spectral densities. The next step is the random process generation with its first continuous derivative. The concatenation procedure was proposed to create the continuous function reconstruction with the pointwise specification (local maxima and minima sequence). The regression analysis of two random variables was used to evaluate the frequency component. Spectral analysis of such a continuous process results in the spectral density function, which finds its application in some methods of random fatigue estimation. The major trait of the proposed method is the coincidence of the rain-flow cycle counting characteristics of the initial and resulting processes.

UQpy: A general purpose Python package and development environment for uncertainty quantification

This paper presents the UQpy software toolbox, an open-source Python package for general uncertainty quantification (UQ) in mathematical and physical systems. The software serves as both a user-ready toolbox that includes many of the latest methods for UQ in computational modeling and a convenient development environment for Python programmers advancing the field of UQ. The paper presents an introduction to the software's architecture and existing capabilities, divided in the code in a set of modules centered around different UQ tasks such as sampling methods, generation of random processes and random fields, probabilistic inverse modeling, reliability analysis, surrogate modeling, and active learning. The paper also highlights the importance of the RunModel module, which is used to drive simulations in the uncertainty analyses performed in UQpy. This module conveniently allows the user to define computational models directly in Python, or to run simulations from a third-party software in serial or in parallel. To illustrate the various capabilities, two examples are tracked throughout the paper and analyzed repeatedly for various UQ tasks. The first is a Python model solving a nonlinear structural dynamics problem, used to illustrate UQpy's capabilities in sampling and forward propagation of high dimensional random vectors (stochastic processes), and probabilistic inference. The second model is a third-party Abaqus finite element model solving the thermomechanical response of a beam structure. This example is used to illustrate UQpy's capabilities in variance reduction sampling techniques, reliability analysis, surrogate modeling and active learning techniques.

Nonlinear Approximation in the Large Deviations Principle

The Markov random processes and their approximations are considered. The main object of study is the exponential generator of random processes with independent increments, which are the solution of the problems of large deviations. These processes satisfy the conditions that make it possible to consider the Poisson and Levy approximation. Generators of random processes are normalized by nonlinear parameters. Found an explicit form of normalization parameter estimation.

On spectral representation method and Karhunen–Loève expansion in modelling construction material properties

Randomness in construction material properties (e.g. Young's modulus) can be simulated by stationary random processes or random fields. To check the stationarity of commonly used techniques, three random process generation methods were considered: Xn(t), Yn(t), and Zn(t). Methods Xn(t) and Yn(t) are based on a truncation of the spectral representation method with the first n terms. Xn(t) has random amplitudes while Yn(t) has random harmonics phases. Method Zn(t) is based on the Karhunen–Loève expansion with the first n terms as well. The effects of the truncation technique on the mean-square error, covariance function, and scale of fluctuation were examined in this study; these three methods were shown to have biased estimations of variance with finite n. Modified forms for those methods were proposed to ensure the truncated processes were still zero-mean, unit-variance, and had a controllable scale of fluctuation; in particular, the modified form of Karhunen–Loève expansion was shown to be stationary in variance. As a result, the modified forms for those three methods are advantageous in simulating statistically homogenous material properties. The effectiveness of the modified forms was demonstrated by a numerical example.

ГЕНЕРУВАННЯ ВИПАДКОВИХ ПРОЦЕСІВ ДИНАМІЧНИМИ СИСТЕМАМИ

Багато системи представляються за допомогою диференціального рівняння, що зв'язує вхідний сигнал з вихідним. Імпульсна характеристика h(t, u) є просто розв’язок, коли вхідним сигналом служить імпульс в момент часу u. У даній статті розглянуті методи реалізації системи, що описується диференціальним рівнянням n-ого порядку з постійними коефіцієнтами. Будь-яке несингулярне лінійне перетворення вектора стану веде до нового уявлення стану. Отримано дисперсійне рівняння, що не містить прийнятого сигналу і є матричним рівнянням Ріккаті.

МЕТОД ГЕНЕРИРОВАНИЯ ДВУМЕРНОГО БИНАРНОГО СЛУЧАЙНОГО ПРОЦЕССА

The PURPOSE of the study is search for new and modernization of known generation methods of random processes with specified static (probability distribution law) and dynamic (correlation function) probabilistic properties. METHODS. The main research methods used in the study are the probability theory, mathematical statistics and numerical methods. RESULTS. The object of the study is a permutational technology of random process generation with a simultaneously specified probability distribution law and an autocorrelation function. This technology is simple in software implementation, has high response speed and allows to introduce the required probability properties with the accuracy of their reflection, sufficient for engineering applications (e.g. in simulation modeling). The Bernoulli (binary) law is taken as a law of probability distribution of the generated process. The conducted analysis of the probabilistic properties of the behavior of the permutation procedure components has resulted in the identification of new, previously unknown functional capabilities of this method of random process generation. Probabilistic properties (probability distribution laws and autocorrelation functions of the candidate vector components) are systematized depending on a mathematically justified permutation variant. The study is given to two variants of the permutational method that allows to generate a periodically correlated binary process. It is proved that these variants simultaneously provide the possibility to generate a two-dimensional binary random process. CONCLUSION. Conducted theoretical studies allowed to expand the functionalities of the permutational procedure of binary random process generation due to the possibility to generate a two-dimensional random binary process.

On the numerical generation of positive-axis-defined distributions with an exponential autocorrelation function

Stochastic modeling commonly requires random process generation with an exponential autocorrelation function (ACF). These random processes may be represented as a solution of a stochastic differential equation (SDE) of the first order and usually have one-sided (positive-axis-defined) distributions. However, adoption of the SDE-based method faces serious limitations due to difficulties with the numerical solution. To overcome this issue we propose a tractable general numerical solution of the above-mentioned SDE that preserves solution positivity and accuracy, and validate it with numerical simulations.

The Behavior of the Generator Normalization Factor in Approximation of Random Processes

Markov random evolutions and their approximations are analyzed. The main object of study is generators of random processes with independent increments. These processes are considered in Poisson approximation and Levy approximation schemes. Generators of random processes are normalized by parameters that are nonlinear functions. The explicit form of such normalization parameters is shown. The asymptotic representation of generators in both approximation schemes is shown. Normalizing factors of random evolution are presented.

Discrete-space time-fractional processes

Abstract A time-fractional diffusion process defined in a discrete probability setting is studied. Working in continuous time, the infinitesimal generators of random processes are discretized and the diffusion equation generalized by allowing the time derivative to be fractional, i.e. of non-integer order. The properties of the resulting distributions are studied in terms of the Mittag-Leffler function. We discuss the computation of these distribution functions by deriving new global rational approximations for the Mittag-Leffler function that account for both its initial Taylor series and asymptotic power-law tail behaviours. Furthermore, we derive integral representations for both the continuous and the discrete time-fractional distributions and use these to prove a convergence theorem.

Generation of random processes for fatigue testing

In fatigue testing there is a need to generate multivariate random processes meeting certain conditions on distributions, crossing properties and regularity. In this paper it is demonstrated how these conditions can be satisfied. The required process is generated by a random time deformation working on a Gaussian process.